# Properties

 Label 2-12e3-12.11-c3-0-29 Degree $2$ Conductor $1728$ Sign $-1$ Analytic cond. $101.955$ Root an. cond. $10.0972$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 20.8i·5-s + 13.9i·7-s + 34.5·11-s + 31.3·13-s + 34.4i·17-s + 120. i·19-s − 137.·23-s − 309.·25-s + 93.1i·29-s + 111. i·31-s − 289.·35-s + 146.·37-s − 8.44i·41-s + 427. i·43-s + 318.·47-s + ⋯
 L(s)  = 1 + 1.86i·5-s + 0.750i·7-s + 0.945·11-s + 0.668·13-s + 0.491i·17-s + 1.45i·19-s − 1.24·23-s − 2.47·25-s + 0.596i·29-s + 0.645i·31-s − 1.40·35-s + 0.651·37-s − 0.0321i·41-s + 1.51i·43-s + 0.989·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $-1$ Analytic conductor: $$101.955$$ Root analytic conductor: $$10.0972$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1727, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.025399803$$ $$L(\frac12)$$ $$\approx$$ $$2.025399803$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 - 20.8iT - 125T^{2}$$
7 $$1 - 13.9iT - 343T^{2}$$
11 $$1 - 34.5T + 1.33e3T^{2}$$
13 $$1 - 31.3T + 2.19e3T^{2}$$
17 $$1 - 34.4iT - 4.91e3T^{2}$$
19 $$1 - 120. iT - 6.85e3T^{2}$$
23 $$1 + 137.T + 1.21e4T^{2}$$
29 $$1 - 93.1iT - 2.43e4T^{2}$$
31 $$1 - 111. iT - 2.97e4T^{2}$$
37 $$1 - 146.T + 5.06e4T^{2}$$
41 $$1 + 8.44iT - 6.89e4T^{2}$$
43 $$1 - 427. iT - 7.95e4T^{2}$$
47 $$1 - 318.T + 1.03e5T^{2}$$
53 $$1 + 291. iT - 1.48e5T^{2}$$
59 $$1 - 364.T + 2.05e5T^{2}$$
61 $$1 - 289.T + 2.26e5T^{2}$$
67 $$1 - 305. iT - 3.00e5T^{2}$$
71 $$1 - 102.T + 3.57e5T^{2}$$
73 $$1 - 442.T + 3.89e5T^{2}$$
79 $$1 + 245. iT - 4.93e5T^{2}$$
83 $$1 + 478.T + 5.71e5T^{2}$$
89 $$1 + 1.41e3iT - 7.04e5T^{2}$$
97 $$1 - 1.15e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$